Linear Operators: General theory |
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Page 183
... follows immediately . Clearly , g ( p ( ) ) is a μ - integrable simple function if g is a 2- integrable simple function . From the definition of μ , it follows that for such a function g we have √ 2 5 ( 82 ) μ2 ( d82 ) = √6−1 ( E ) ...
... follows immediately . Clearly , g ( p ( ) ) is a μ - integrable simple function if g is a 2- integrable simple function . From the definition of μ , it follows that for such a function g we have √ 2 5 ( 82 ) μ2 ( d82 ) = √6−1 ( E ) ...
Page 403
... follows easily from ( 1 ) that ( 2 ) μs , ( e§1 ) = μs 。( e ) . Equation ( 2 ) enables us to define a Gauss measure in real Hilbert space as follows . Call a Borel subset e of H a cylinder set if there exists an orthogonal projection E ...
... follows easily from ( 1 ) that ( 2 ) μs , ( e§1 ) = μs 。( e ) . Equation ( 2 ) enables us to define a Gauss measure in real Hilbert space as follows . Call a Borel subset e of H a cylinder set if there exists an orthogonal projection E ...
Page 689
... follows ( IV.8.9 ) that the set { A ( x ) f , 0 ≤ x } is weakly sequentially compact in L1 ( S , E , u ) . The conclusion now . follows from Theorem 1. Q.E.D. ∞ = The remaining part of the section will be concerned exclusively with ...
... follows ( IV.8.9 ) that the set { A ( x ) f , 0 ≤ x } is weakly sequentially compact in L1 ( S , E , u ) . The conclusion now . follows from Theorem 1. Q.E.D. ∞ = The remaining part of the section will be concerned exclusively with ...
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A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ